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We know that all objects with mass exert forces on all other objects of mass such that

$$ F = \frac{GMm}{R^2}.$$

And as others have discussed the planets do interfere with each other gravitationally to a small degree.

My question is how reliant the solar system is on its exact structure. If a planet were to change its alignment or orbit or gravitational effect on other planets, through gain of a mass through an asteroidal collision for example.

Would a deviation in the structure of the solar system as it is cause it to collapse? e.g planets change orbits significantly enough to drift away from the sun or drift into it?

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    Good question, but it goes deep into orbital resonances. The familiar rules of gravitation start to look extremely different when played out over millions of years. I could confidently say it wouldn't cause planets to plummet to their death in the very next orbit, but after that the system is quite chaotic. – Alan Rominger May 14 '13 at 12:04
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    I second Alan's comment; over a long enough time period, all the planets will either be ejected from the solar system, or collide with the sun. However, removing one of the planets at the present time probably won't have much of an effect in the near term. – Dmitry Brant May 14 '13 at 13:46
  • I doubt that Newtonian gravity is logically self-consistent if you don't have conservation of mass. Certainly GR isn't. That means you can't just have a planet disappear. A better way to phrase the question would be that the planet is rapidly accelerated away by some external force. –  May 14 '13 at 15:00
  • @BenCrowell i deliberately avoided that because then people would ask what caused it, which direction is it travelling, if it overlaps any other orbits etc, and those arent the kind of answers i want, the kind im after are ones which focus on the lack of planets resultant effect on the rest of the planets orbits. hence why ceasing to exist seemed the best choice. –  May 14 '13 at 15:24
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    also @downvoters, would you care to comment? I cant improve the question if you don't tell me whats wrong! –  May 14 '13 at 15:25
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    @BenCrowell there is no reason to constrain the problem as such. There is no lack of 'self-consistency', the problem just becomes one of artificial initial conditions---i.e. the current positions and velocities, but the dynamic situation has one fewer body. RhysW's response is exactly correct, if you required a 'rapid acceleration' you might as-well require an explanation for it, or an entire universe where such an explanation would naturally arise. – DilithiumMatrix May 14 '13 at 16:00
  • @zhermes: So are you claiming that Newtonian gravity is self-consistent without conservation of mass, but its relativistic extension, GR, isn't? I suppose that's possible, since Newtonian gravity isn't a field theory like GR and electromagnetism. For both GR and electromagnetism, there is a continuity equation that is required for self-consistency. –  May 14 '13 at 16:20
  • @RhysW: I downvoted simply because I think the question is kind of silly and unmotivated. The way it's stated also violates both conservation of mass and conservation of energy. –  May 14 '13 at 16:21
  • @zhermes: "if you required a 'rapid acceleration' you might as-well require an explanation for it, or an entire universe where such an explanation would naturally arise." There is a difference between a hypothetical force that is consistent with the laws of physics and a hypothetical process that violates conservation of mass and energy. –  May 14 '13 at 16:29
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    @BenCrowell what I'm saying is that there is no reason to require conservation of mass in this situation. Say you have a mass on a spring, with some initial oscillation. Then say half the mass disappears. There's no issue: it's the same as saying you have half the initial mass with some artificial initial displacement and velocity. – DilithiumMatrix May 14 '13 at 17:13
  • @zhermes Before this becomes a long discussion, I suggest you move this to [chat] if you are planning on continuing (create a new room). Thanks :) – Manishearth May 14 '13 at 17:17
  • @BenCrowell See above^^ – Manishearth May 14 '13 at 17:17
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    Related: http://physics.stackexchange.com/q/34600/ http://physics.stackexchange.com/q/5456/ My personal temptation is to close this with extreme prejudice until the asker can state what they want to know instead of presupposing the impossible. But that's just me. – dmckee --- ex-moderator kitten May 14 '13 at 19:46
  • Googling on "lyapunov exponent solar system" (without the quotes) gives a whole bunch of relevant papers. –  May 14 '13 at 21:22

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This is something I played with while testing a n-body code I wrote during college. Unfortunately I don't have any animations, or even the original code anymore - but I can report qualitative results.

Removing Jupiter and Saturn does indeed have a significant destabilizing effect -- an a chaotic one at that (i.e. depending on precise initial conditions, and varying on numerical accuracy) -- leading to the dynamical instability of numerous planets.

Removing the other planets had no effect on dynamical stability, but there were some small changes to periods, etc.

This result should be expected as the gravitational effects of planets other than Jupiter (and saturn to a lesser degree) are almost entirely negligible on the dynamics of other planets.

DilithiumMatrix
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